RIYADH, May 13 (Reuters) - Saudi Arabia has invited Iran's foreign minister to visit, Riyadh said on Tuesday, hinting at the possibility of a thaw between two bitter rivals, whose struggle for influence is evident in conflicts throughout the region.

Iranian President Hassan Rouhani has adopted a conciliatory tone towards Tehran's neighbours since taking office last year, but while Foreign Minister Mohammad Javad Zarif has visited other Gulf Arab states, he has not yet been to Saudi Arabia.

Rapprochement between the two countries would have ramifications across the Middle East, potentially cooling political and military struggles in Syria, Iraq, Lebanon, Bahrain and Yemen.

However, with Riyadh and Tehran giving full-throated backing to opposing sides in Syria's civil war, and accusing each other of fuelling the bloodshed, the prospects for any meaningful detente now appear slim, analysts say.

Foreign Minister Prince Saud al-Faisal told a news conference that Zarif had been given an invitation to the kingdom but that despite Iran's past declarations of a wish to improve ties, the visit had not transpired. He did not say when Riyadh issued the invitation or if Iran had formally responded.

"Any time that (Zarif) sees fit to come, we are willing to receive him. Iran is a neighbour, we have relations with them and we will negotiate with them, we will talk with them," he said.

Sunni Muslim Saudi Arabia and Shi'ite Iran have long supported competing factions in Arab countries, often along sectarian lines. But Iranian backing for Syrian President Bashar al-Assad, and the aid Riyadh has given to rebels trying to oust him, has raised their mutual hostility to unprecedented levels.

Saudi Arabia accuses Iran of fomenting unrest among the Shi'ite majority in its neighbour Bahrain, and the sect's minority in its own Eastern Province, and also charges Tehran with plotting to assassinate its envoy in Washington in 2011.

Iran denies those accusations, as well as Saudi suspicions, shared with Western powers, that it has been using its declared civilian nuclear energy programme as a front to covertly develop an atomic bomb capability.

SUSPICIONS

But since taking office in August, the moderate Rouhani has overseen a conciliatory shift in Iran's hitherto confrontational foreign relations. The most tangible result so far was Iran's Nov. 24 interim nuclear deal with global powers.

Although Iran's president has a big voice in determining Tehran's foreign policy, the ultimate say is in the hands of clerical Supreme Leader Ayatollah Ali Khamenei.

"It's only a matter of time before Zarif takes up the invitation and goes to Riyadh. It's a question of coordination at home with the leader. But it's inevitable that he go and important that he does," said Anoush Ehteshami, director of the al-Sabah programme for international relations at Durham University in Britain.

"The Saudis are calling his bluff and saying 'come'."

Saudi officials have remained suspicious, however, and have accused Iran of being "an occupying power" in Syria, where they describe Assad as carrying out genocide against the country's civilian population via air strikes in urban areas.

"Our hope is that Iran becomes part of the effort to make the region as safe and as prosperous as possible and not become part of the problem," the Saudi foreign minister said.

Suspicion between the two is deeply rooted, with Saudi Arabia's ruling princes worried that Iran's clerical elite remains determined to export the message of its 1979 Islamic Revolution to Shi'ites across the Middle East.

Iranian leaders regard Riyadh as a stooge for their American foes, and remain angry at the Saudi role in backing Iraq during its eight-year war with Iran.

"It's not a rapprochement. All the issues are still there, (Iran's) interference that we have seen, all of it will come again on to the table. But it's better to meet your counterpart and to see the margin for compromise," said Abdulaziz al-Sager, head of the Gulf Research Centre, based in Jeddah and Geneva.

Formal visits by officials of each country to the other remain rare, but Saudi Arabia's position as birthplace of Islam and guardian of its most sacred places has sometimes allowed Iranian leaders to conduct discrete diplomacy during pilgrimage.

Rouhani's predecessor Mahmoud Ahmadinejad, who was regarded by Riyadh as a source of much of the tension between the countries, did meet Saudi King Abdullah at an Organisation of Islamic Cooperation summit in Mecca in 2012.

Abdullah placed Ahmadinejad at his right hand side while receiving leaders of other Muslim countries in an apparently emollient gesture aimed at showing Saudi Arabia wanted to reduce tensions with Iran and sectarian divisions in the region.

After Ahmadinejad's foreign minister, Manouchehr Mottaki, visited Riyadh in 2009 in an effort to lessen tensions, King Abdullah told U.S. officials he had warned the Iranians that "you as Persians have no business meddling in Arab matters", according to a U.S. diplomatic cable released by WikiLeaks.

"Iran's goal is to cause problems ... There is no doubt something unstable about them," the Saudi monarch told visiting U.S. counter-terrorism adviser John Brennan, according to the cable.

(Additional reporting by William Maclean and Yara Bayoumy; Editing by Sami Aboudi and Mark Heinrich)